The problem is a type of "Packing Problem". Regarding polyominoes we are interested in tiling specific regions.

It is known that the problem of answering whether a specific region can be filled with tiles is undecidable. And that finding a solution, in general, is NP-Complete. So we can never do much better than trying all possible configurations - i.e. pick something and exhaustively show that it doesn't work, backtrack and try again.

As to the "simpler" problems like L-trominoes, it is essentially the same problem.

So, to answer directly: there is no easy way to determine whether there is a solution. Just as there is no easy solution for the travelling salesman problem, the tiling problem, or the boolean satisfiability problem (SAT) (on a personal note, I immediately think of this problem, and could fairly easily cast the polyomino problem as a SAT problem).

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Be Bruijn's Theorem

The problem is a type of "Packing Problem". Regarding polyominoes we are interested in tiling specific regions.

It is known that the problem of answering whether a specific region can be filled with tiles is undecidable. And that finding a solution, in general, is NP-Complete. So we can never do much better than trying all possible configurations - i.e. pick something and exhaustively show that it doesn't work, backtrack and try again.

As to the "simpler" problems like L-trominoes, it is essentially the same problem.

So, to answer directly: there is no easy way to determine whether there is a solution. Just as there is no easy solution for the travelling salesman problem, the tiling problem, or the boolean satisfiability problem (SAT) (on a personal note, I immediately think of this problem, and could fairly easily cast the polyomino problem as a SAT problem).

Related:
More Papers
Be Bruijn's Theorem